Reformulating this in our framework, we find that the value function \(v\)  is a sufficiency measure. To see this, let each \(\mathbf {z} \sim \mathcal {D}\)  be a sample in which a random subset of variables \(S\)  are held at their original values, while remaining features \(R\)  are drawn from a fixed distribution \(\mathcal {D}(\cdot |S)\) .The diversity of Shapley value algorithms is largely due to variation in how this distribution is defined. Popular choices include the marginal \(P(\mathbf {X}^R)\)  [1]}; conditional \(P(\mathbf {X}^R|\mathbf {x}^S)\)  [2]}; and interventional \(P(\mathbf {X}^R|do(\mathbf {x}^S))\)  [3]} distributions.
